K-Theory and Normal Complexes

• Applicant: Professor Dr. Christian Haase
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539974215

Combinatorial intersection theory is by now established and active area of research which connects algebraic geometry with combinatorics. On one hand, developing combinatorial tools to study algebraic varieties is extremely important as disproportionately large share of the spaces arising in algebraic geometry (as well as in commutative algebra, representation theory and mathematical physics) have an explicit combinatorial structure. On the other hand, since Stanley’s proof of McMullen’s g-conjecture, it was evident that algebro-geometric tools have enormous impact on combinatorics. Recently this was further supported by the groundbraking resolution of Heron-Rota-Welsh log-concavity conjecture by Adiprasito-Huh-Katz. The main breakthrough of their work was to translate techniques of intersection theory to the combinatorial setting of matroids. K-ring of algebraic variety is a refined version of intersection theory. Similar to the intersection theory, it is actively applied to combinatorial problems. However, the situation with K-theory is at present similar to the situation with intersection theory before Adiprasito-Huh-Katz - the scope of application is bounded to the combinatorial problems which admit tools are bounded to the combinatorial problems admitting an algebro-geometric model. We propose to complete the picture and to generalize K-theory construction from algebraic varieties to combinatorial setting. Two starting points of our proposal are the theory of Pukhlikov-Khovanskii algebras which model intersection theory and K-rings for a large class of algebraic varieties and the theory of normal complexes, which generalizes the theory of convex polytopes in setting of intersection theory for matroids. We anticipate that we will be able to achieve our goal by combining these two approaches.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2458: Combinatorial Synergies
• International Connection: Switzerland
• Partner Organisation: Schweizerischer Nationalfonds (SNF)
• Cooperation Partner: Dr. Leonid Monin